Zig-zagging in geometrical reasoning in technological collaborative environments: a Mathematical Working Space-framed study concerning cognition and affect

Abstract

This study highlights the importance of cognition-affect interaction pathways in the construction of mathematical knowledge. Scientific output demands further research on the conceptual structure underlying such interaction aimed at coping with the high complexity of its interpretation. The paper discusses the effectiveness of using a dynamic model such as that outlined in the Mathematical Working Spaces (MWS) framework, in order to describe the interplay between cognition and affect in the transitions from instrumental to discursive geneses in geometrical reasoning. The results based on empirical data from a teaching experiment at a middle school show that the use of dynamic geometry software favours students’ attitudinal and volitional dimensions and helps them to maintain productive affective pathways, affording greater intellectual independence in mathematical work and interaction with the context that impact learning opportunities in geometric proofs. The reflective and heuristic dimensions of teacher mediation in students’ learning is crucial in the transition from instrumental to discursive genesis and working stability in the Instrumental-Discursive plane of MWS.

Appendices

Appendix 1

Students´ responses to task 5 of the GeoGebra sequence serve to illustrate different levels of empirical proof, according to Gutiérrez’ (2005) classification.

Task 5: The Vizier suggests that you tile the floor of the “Chamber of Abencerrajes” (this room is the sultan’s bedroom in the Alhambra of Granada) using tiles with equal sides, in order to save money (these tiles are cheaper than those with irregular shapes). What shapes can these tiles have? Can you obtain more regular tessellations using regular tiles with more sides? Justify your answers.

• Naïve experiment: Verification on the property is based on the examples that students have generated. An example is Fig. 7:

  Fig. 7

  

  Naïve experiment

• Crucial experiment: Students considered their examples as class-representative. They made measurements in order to prove that only those regular polygons whose sides sum up to exactly 360° can tile the plane (Fig. 8).

  Fig. 8

  

  Crucial experiment

• “Exemplified” is when the proof consists only of showing the existence of a crucial example. The response is shown in Fig. 8.

• “Constructive” is when the proof affects the way the example is obtained. Students included, besides the above image, a written response where they claimed that, because they are regular polygons, all their sides are equal, and, since they were constructed by means of isometries, the angles are preserved in the polygons obtained that way. Since the construction process guarantees equal angles meeting at each vertex of the tessellation, they could already affirm that only in three cases do they sum up to exactly 360°.

• “Analytical” is when the proof is based on empirically observed mathematical properties. Students included, besides Fig. 8, a written response where they claimed that equal angles meet at each vertex of the tessellation. In the case of the triangle, six angles of 60° each; in the case of qualateral, four angles of 90° each, in the case of the pentagon, four angles 180° each; in the case of the hexagon, three angles of 120° each; and from then on, they verified that the angles of the regular polygons will always measure more than 120°, and therefore, three of them will never sum up to 360°.

• “Intellectual” is when the proof is based on accepted mathematical properties and deductive relationships between elements in the example. Students included, besides Fig. 8, a written response with the value of the angles of regular polygons up to 12 sides, obtained from the formula:

$$\propto_{\textit{n}}\,= \frac{{180^\circ \cdot({\textit{n}} - 2)}}{\textit{n}}$$

They verified that the angle values increase as the number of sides increase. They affirmed that, since the sum of the angles meeting at each vertex must be 360º, from the hexagon onward, only three polygons can meet to form a tesselation, and they verified, with the values obtained from the formula, that in any case α _n · 3 = 360^o para 6 < n ≤ 360^o.

• Generic example: being aware of the need to generalise, students selected a class-representative example. The proof entails abstract reasoning that refers to general class properties and elements that are obtained by operating with or transforming the example. In this case, the proofs are not confined to the reflection of the empirical activity, but transform it into references to abstract class properties and the deductive reasoning that links them.

• “Exemplified” is when the proof consists only of showing the existence of a crucial example. Students added to Fig. 8a written response, in which they claimed, based on the examples, that the only regular tessellations correspond to the three polygons whose inner angle is a divisor of 360°.

• “Constructive” is when the proof affects the way the example is obtained. Students add to Fig. 8 a written response, in which they claimed that the property is valid for any regular triangle, square or regular hexagon because the way they are constructed guarantees the invariability of the value of their inner angle. Besides, they argued that there only exist three such regular tessellations because the inner angle must be a divisor of 360 and there are no more divisors of 360 which correspond to the value of the inner angle of a regular polygon. The next divisor is 180, which is not a possible value for the angle of a polygon.

• “Analytical” is when the proof is based on empirically observed mathematical properties. Figure 8 allows the students to observe that the three regular tessellations are made up of polygons whose inner angles are divisors of 360. Then, they calculated all the divisors of 360 and the value of the inner angles of many polygons with more than six sides, verifying that they form an increasing sequence, and they checked that there are no more divisors of 360 after 120 which can be angles of a regular polygon. The remaining divisors are 180 and 360 and they are not possible values of the inner angle of a polygon.

• “Intellectual” is when the proof is based on accepted mathematical properties and deductive relationships between elements in the example. As in the former case, students obtained all the divisors of 360. Afterwards, they studied the sequence which allowed them to obtain the value of the inner angles of a regular polygon with n sides $( \propto_{\text{n}} ) = \frac{{180^\circ \cdot({\text{n}} - 2)}}{\text{n}}$. They verified that it is an increasing sequence whose bounds are 60º and 180º, its upper bound being 180º. Therefore, there is no other regular polygon whose number of sides is more than six and its inner angle corresponds to a divisor of 360º greater than 120º. Consequently, no other regular polygon can be used to build a tessellation. In this way, students proved both the existence and the unicity.

Appendix 2

See Table 3.

Table 3 Grid of mathematical attitudes